Question

Let \(\eta \in \mathcal{C}^{\infty}\left(\mathbb{R}^{n}\right)\) be a smooth function on \(\mathbb{R}^{n}\), and let \(\varphi\) be a distribution. Show that \(\eta \varphi\) is also a distribution. What is the natural definition for \(\eta \varphi ?\) What is \((\eta \varphi)^{\prime}\), the derivative of \(\eta \varphi\) ?

Short answer

The product \( \eta\varphi \) of a smooth function \( \eta \) and a distribution \( \varphi \) is also a distribution by definition, since it is a continuous linear functional from \( \mathcal{C}^\infty \) to \( \mathbb{R} \). The natural definition is \((\eta\varphi)(\psi) = \varphi(\eta\psi)\) for all \( \psi \in \mathcal{C}^\infty\). The derivative of this multiplication is defined as \((\eta\varphi)' = \eta'\varphi + \eta\varphi' \).

Step-by-step solution

Define the multiplication of a function and distribution

The multiplication of a test function \( \eta \) and a distribution \( \varphi \) can be defined as \((\eta \varphi)(\psi) = \varphi(\eta \psi)\) for all \( \psi \in \mathcal{C}^\infty\). The product \( \eta\varphi \) is a linear functional from \( \mathcal{C}^\infty \) to \( \mathbb{R} \).

Check the continuity of the multiplication

Next it needs to be confirmed whether this multiplication is continuous or not. For \( \mathcal{C}^\infty \), we have the property that if \( \psi_n \rightarrow \psi \) in \( \mathcal{C}^\infty \), then \( \eta\psi_n \rightarrow \eta\psi \) which means the product \( \eta\varphi \) is continuous given that \( \varphi \) is a distribution. Thus, according to definition, \( \eta\varphi \) is also a distribution.

Define the derivative of the multiplication

The derivative \((\eta\varphi)'\) of the multiplication \( \eta\varphi \) can be defined using the rule of distributional derivatives. We have \((\eta\varphi)' = \eta'\varphi + \eta\varphi' \).

Key concepts

Smooth Functions

In the realm of mathematics, especially in analysis, smooth functions are a cornerstone. A smooth function is one that has derivatives of all orders. If \[ \eta \in \mathcal{C}^{\infty}(\mathbb{R}^{n}) \]this means that \( \eta \)is infinitely differentiable—making it incredibly adaptable and flexible. These functions play a vital role in distribution theory because they help facilitate interactions between different mathematical objects, like distributions.
Imagine a smooth function like a perfectly stretchy fabric: it can be pulled and manipulated without tearing or breaking. This characteristic makes it an essential tool in various mathematical operations, like multiplying with distributions. Being infinitely differentiable also ensures that the technical manipulations remain valid across all mathematical operations. Smooth functions ensure these processes occur seamlessly, maintaining order and continuity.

Distribution Continuity

When dealing with distributions, continuity is not about smooth lines. Instead, it's about maintaining stability when applied to sequences of test functions. A distribution \( \varphi \)is a linear functional that assigns numbers to smooth functions in a consistent way. More technically, if \( \psi_n \rightarrow \psi \)in the space of test functions, then the transformed sequence maintains its limit.
In simpler terms, if you start with a series of functions converging to a specific function, applying a distribution should lead to results that also show convergence. This idea of continuity ensures that when you multiply a distribution by a smooth function, the resulting product, \( \eta \varphi \)is still a distribution. This confirms that operations involving distributions will behave consistently, without unexpected jumps or breaks. It means you can trust the mathematical "fabric" you're working with.

Distributional Derivatives

The notion of derivatives is extended into distributions using the concept of distributional derivatives. In classical calculus, the derivative of a function gives us the rate of change. But, for distributions, we approach derivatives differently.
For a product of a smooth function and a distribution, \( (\eta \varphi)' \)is defined by the formula:

• \( (\eta \varphi)' = \eta' \varphi + \eta \varphi' \)

This formula is comprised of two parts:

• \( \eta' \varphi \): Differentiating the smooth function and then multiplying by the distribution.
• \( \eta \varphi' \): Keeping the smooth function constant while differentiating the distribution.

This setup allows us to explore the rate of change even when dealing with more complex structures like distributions. Understanding this framework is crucial because it highlights how distributions behave under differentiation, preserving the consistency of mathematical operations even in this more generalized context.